|The Games and Puzzles Journal Issue 35, September-October 2004|
Editor's Note: This article has been delayed in publication mainly because I was hoping to find a better way of presenting the dissection diagrams. Mr Willcocks provided A4-size diagrams prepared for him by Dr J. D. Skinner, but it was not possible to reproduce them adequately on the screen. However, George Bell (19 December 2004) has now been able to provide a scale drawing gif-image of the first dissection (A1) as shown above. The impossibility of showing an accurate diagram is evident from the fact that the square side is 6732 units and the screen width is only 800 pixels. So the above image, which is 600 pixels wide, compresses 11.22 units to a pixel. So squares smaller than this, as occur in B1, B3, C1, C2 below, would be lost. For the diagrams in the text I have used the conventional method of showing the squares as rectangles.
During the past 75 years much work has been done on the problem of dividing a rectangle into smaller squares all of different sizes. When the rectangle itself is a square the problem is known as Squaring the Square and a comprehensive theory was developed by four mathematicians at Cambridge by which they showed a relationship between the elements of dissections and the currents in certain electrical networks. Details may be found in  and . A very interesting non-mathematical but authoritative account may be found in . The following notes are based entirely on the electrical theory.
In the diagram of Fig.(1a) ABC represents a network where for simplicity interior wiring has been omitted. DEF represents the map which is the dual of an equivalent map to ABC (interior wiring again omitted). These two networks are joined by the wires BE, BD, DC, CF, and FA. If now a current enters the network at A and leaves at E, the theory tells us that (1) the network gives rise to a squared square and (2) the wires BD and CF carry zero currents.
Lying in my bath one evening with a mental picture of such a network, I realised that the wire XD could equally well be drawn as XB and similarly the wire YB could be drawn as YD. What I had done was to replace two sides of the quadrilateral BYDX with the other two sides. It seemed that the natural thing to do was to replace the diagonal BD with the other diagonal XY and see what resulted. I had no reason to expect any interesting outcome, but on evaluating the new network I discovered that (1) I had a squared square, (2) the currents in BX and DY were numerically equal, and (3) vectors could be assigned as shown in Fig.(1b). Knowledge of (2) and (3) can simplify the calculation of currents in the network.
Experiment showed that a similar procedure could be followed with the wire CF or with both wires with a view to obtaining a single pair of equal squares in two different ways or two pairs of equal squares in one way. I have constructed a considerable number of such squares without finding a failure, but I lack a strict proof justifying the procedure. As with all other methods of this kind it is important to check that there is no unwanted imperfection.
An example of three squares constructed in this way showing an interesting arithmetical relationship is given in the accompanying diagrams C1, C2, C3. The sum of the duplicated squares in C1 and C2 (92 + 208) is equal to the sum of the two duplicated pairs in C3 (97 + 203).
This method is not restricted to the use of self-dual maps. A dissection of a rectangle in two different ways but with a corner square in one equal to a corner square in the other will yield two different tripolar networks and one of these can be used with the dual of the other.
In this way we can form four networks, and it is their close relationships with each other which result in some curious arithmetical equalities, such as that noted above. From A and B we have 431 + 187 = 332 + 286.
The writer does not claim that all initial networks give such interesting results. He has been content to show some that he has found.
A1: 34 squares, side 6732 units.
A2: 34 squares, side 6876 units.
A3: 35 squares, side 9447 units.
B1: 34 squares, side 6705 units.
B2: 34 squares, side 6732 units.
B3: 35 squares, side 4669 units.
C1: 34 squares, side 6705 units.
C2: 34 squares, side 6876 units.
C3: 35 squares, side 9413 units.
It is possible to make further speculations. It is obvious that the process is reversible. The method employed results in all cases with a square containing 3 component squares situated as in Fig.(2a). Reversing the process the two squares A are removed and a zero current is introduced.
What happens if we remove by this process in a square with a configuration as in Fig.(2b) the two squares A which adjoin square D? The small number of examples I have tested have all yielded squared squares.
The sceptical reader might care to test the inverse method with the following square, Fig.(3). At first sight this does not look like a candidate but it can be made one by treating the square  as being 4 squares of side 2 surrounding a zero square. To my surprise I found that this resulted in a square (imperfect of side 313). Is it conceivable that a perfect square could be found in this way?
Fig.(3). 21 squares, side 107
In conclusion I would like to express my thanks to Dr J. D. Skinner for his help with the diagrams.
 The dissection of rectangles into squares R. L. Brooks, C. A. B. Smith, A. H. Stone, W. T. Tutte, (Duke Math J. Vol.7 p.312-340)
 Squaring the Square W. T. Tutte (Can. J. Math. Vol.2 p.197-209, 1950).
 Squaring the Square W. T. Tutte (Chapter 17 of the 2nd Scientific American Book of Mathematical Puzzles and Diversions).
|Update (2 June 2013): My more recent computers have had no trouble with access to the site. I have corresponded with the author Stuart Anderson. There is now a page devoted to T. H. Willcocks and his work: THW on www.squaring.net. Mr Willcocks reached the age of 100 on 19 April 2012.|